Advertisements
Advertisements
Question
In the following example verify that the given expression is a solution of the corresponding differential equation:
y = `"e"^"ax"; "x" "dy"/"dx" = "y" log "y"`
Advertisements
Solution
y = `"e"^"ax"`
∴ log y = log `"e"^"ax"` = ax log e
∴ log y = ax .....(1) .....[∵ log e = 1]
Differentiating w.r.t. x, we get
`1/"y" * "dy"/"dx" = "a" xx 1`
∴ `"dy"/"dx" = "ay"`
∴ `"x""dy"/"dx" = ("ax")"y"`
∴ `"x" "dy"/"dx" = "y" log "y"` ....[By (1)]
Hence, y = `"e"^"ax"` is a solution of the D.E. `"x" "dy"/"dx" = "y" log "y"`.
APPEARS IN
RELATED QUESTIONS
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
c1x3 + c2y2 = 5
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
y = e−2x (A cos x + B sin x)
Find the differential equation all parabolas having a length of latus rectum 4a and axis is parallel to the axis.
Form the differential equation of family of lines parallel to the line 2x + 3y + 4 = 0.
Solve the following differential equation:
`"dy"/"dx" = (1 + "y")^2/(1 + "x")^2`
For the following differential equation find the particular solution satisfying the given condition:
3ex tan y dx + (1 + ex) sec2 y dy = 0, when x = 0, y = π.
For the following differential equation find the particular solution satisfying the given condition:
`(e^y + 1) cos x + e^y sin x. dy/dx = 0, "when" x = pi/6,` y = 0
For the following differential equation find the particular solution satisfying the given condition:
`("x" + 1) "dy"/"dx" - 1 = 2"e"^-"y" , "y" = 0`, when x = 1
For the following differential equation find the particular solution satisfying the given condition:
`cos("dy"/"dx") = "a", "a" ∈ "R", "y"(0) = 2`
Reduce the following differential equation to the variable separable form and hence solve:
`"x + y""dy"/"dx" = sec("x"^2 + "y"^2)`
Solve the following differential equation:
(x2 + y2)dx - 2xy dy = 0
Choose the correct option from the given alternatives:
x2 + y2 = a2 is a solution of
Choose the correct option from the given alternatives:
The solution of `("x + y")^2 "dy"/"dx" = 1` is
Choose the correct option from the given alternatives:
The solution of `"dy"/"dx" = ("y" + sqrt("x"^2 - "y"^2))/"x"` is
Obtain the differential equation by eliminating the arbitrary constants from the following equation:
`"y"^2 = "a"("b - x")("b + x")`
In the following example verify that the given function is a solution of the differential equation.
`"y" = 3 "cos" (log "x") + 4 sin (log "x"); "x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`
Form the differential equation of all parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.
Form the differential equation of the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola `"x"^2/16 - "y"^2/36 = "k"`.
Solve the following differential equation:
`"dy"/"dx" = ("2y" - "x")/("2y + x")`
Solve the following differential equation:
x dy = (x + y + 1) dx
Solve the following differential equation:
`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`
Find the particular solution of the following differential equation:
`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1
Find the particular solution of the following differential equation:
y(1 + log x) = (log xx) `"dy"/"dx"`, when y(e) = e2
Select and write the correct alternative from the given option for the question
The solution of `("d"y)/("d"x)` = 1 is
Select and write the correct alternative from the given option for the question
The solutiion of `("d"y)/("d"x) + x^2/y^2` = 0 is
Form the differential equation of y = (c1 + c2)ex
Find the differential equation by eliminating arbitrary constants from the relation y = (c1 + c2x)ex
Find the differential equation from the relation x2 + 4y2 = 4b2
The differential equation having y = (cos-1 x)2 + P (sin-1 x) + Q as its general solution, where P and Q are arbitrary constants, is
Form the differential equation of all straight lines touching the circle x2 + y2 = r2
Find the differential equation of the family of parabolas with vertex at (0, –1) and having axis along the y-axis
Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin
The general solution of the differential equation of all circles having centre at A(- 1, 2) is ______.
The differential equation for all the straight lines which are at the distance of 2 units from the origin is ______.
The differential equation whose solution is (x – h)2 + (y – k)2 = a2 is (where a is a constant) ______.
The differential equation representing the family of ellipse having foci either on the x-axis or on the y-axis centre at the origin and passing through the point (0, 3) is ______.
The differential equation of all circles passing through the origin and having their centres on the X-axis is ______.
The differential equation of all parabolas having vertex at the origin and axis along positive Y-axis is ______.
The differential equation for a2y = log x + b, is ______.
Solve the differential equation
cos2(x – 2y) = `1 - 2dy/dx`
Find the particular solution of the differential equation `x^2 dy/dx + y^2 = xy dy/dx`, if y = 1 when x = 1.
Form the differential equation of all concentric circles having centre at the origin.
